STRONG CONVERGENCE THEOREMS OF RELAXED HYBRID STEEPEST-DESCENT METHODS FOR VARIATIONAL INEQUALITIES

Abstract

Assume that $F$ is a nonlinear operator on a real Hilbert space $H$ which is $\eta$-strongly monotone and $\kappa$-Lipschitzian on a nonempty closed convex subset $C$ of $H$. Assume also that $C$ is the intersection of the fixed point sets of a finite number of nonexpansive mappings on $H$. We develop a relaxed hybrid steepest-descent method which generates an iterative sequence $\{x_n\}$ from an arbitrary initial point $x_0 \in H$. The sequence $\{x_n\}$ is shown to converge in norm to the unique solution $u^*$ of the variational inequality \[ \langle F(u^*), v-u^* \rangle \geq 0 \quad \forall v \in C \] under the conditions which are more general than those in Ref. 19. Applications to constrained generalized pseudoinverse are included.